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Subject:

Re: computing normalized beta weights

From:

Ramapriyan Pratiwadi <[log in to unmask]>

Reply-To:

FSL - FMRIB's Software Library <[log in to unmask]>

Date:

Mon, 23 Aug 2004 21:34:28 +0100

Content-Type:

text/plain

Parts/Attachments:

Parts/Attachments

text/plain (131 lines)

hi,

sorry to regurgitate an old thread, but i don't quite understand the last
post.

for the case of multiple PE's (and COPE's), the dimensions in the formula
sqrt(c'*b*X*X'*b'*c) do not match up; in my case, i have 320 time points and
8 PE's (4 EV's +temporal derivatives).  Therefore, the design matrix is
320x8; if b is a vector, it would have to be 1x320 - how can that be?

i tried transposing the design matrix, to make it 8x320 - then, the
multiplication becomes (1x1)(1x8)(8x320)(320x8)(8x1)(1x1), for each voxel.
However, when the result was divided by the standard deviation of the
filtered_func_data, the values were not normalized between -1 and 1.

normally, we would divide the betas of the PE's by the beta of the constant
column; however, since there is not a constant column, the aforementioned
approached seems suitable.

Thanks in advance,

Ram

-------------------------------------
Brain Behavior Laboratory

University of Pennsylvania
Philadelphia, Pennsylvania, USA


On Tue, 12 Aug 2003 14:39:47 +0100, Tim Behrens <[log in to unmask]> wrote:

>Ok - there are various levels to this answer.
>
>It seems to be a sensible thing to do - effectively you want to know the
>amount amount of the data's standard deviation which is explained by a
>single COPE.
>
>1) If you've only got one PE, then this is relatively trivial.
>the std of the design can be computed easily from design.mat (ascii file
>containing design timeseries). Call this sx. The std of the data is
>
>avwmaths filtered_func_data -Tstd sy
>
>then the fractional deviation explained by your PE is just
>
>avwmaths PE -mul sx -div sy Beta_norm
>
>I think, in this case, you can do this with the unwhitened data as the
>whitening matrix is normalised.
>
>2) If you've got more than 1 PE, life is more complicated (and  I don't
>think you can compute what you want with simple FSL commands ).
>
>You need to project the variance explained by all of your EVs onto a
>single COPE.
>
>if you assume the data is white and demeaned then and your Design is
>demeaned..
>
>the std explained by your cope is:
>
>sqrt(c'*b*X*X'*b'*c)/dof
>
>c is cour contrast, b is your vector of PEs, X is your design, dof is your
>degrees of freedom
>
>if it's not white then
>
>sqrt(c'*b*k*X*X'*k'*b'*c)/dof
>
>k is the whitening matrix.
>(The whitening will change the projection)
>
>and the standard deviation of the whitened data is just std(k*Y)
>
>Dividing one by the other should give you what you want.
>
>Hate to say it, but you might need matlab!!
>
>cheers
>T
>
>
>-------------------------------------------------------------------------------
>Tim Behrens
>Centre for Functional MRI of the Brain
>The John Radcliffe Hospital
>Headley Way Oxford OX3 9DU
>Oxford University
>Work 01865 222782
>Mobile 07980 884537
>-------------------------------------------------------------------------------
>
>On Tue, 12 Aug 2003, Edward Vessel wrote:
>
>> Ok, well, here is something that might give you a feeling for the difference.
>> In a regression with only a single independent variable, the standardized
>> beta equals r (the Pearson's correlation).  This would also be true (I think)
>> if all the variables were totally uncorrelated in a multiple regression.
>>
>> The analysis I have done is one in which I want to look at a correlation
as my
>> statistic, and the standardized Beta coefficient is one way to report a
>> factor loading which takes into account that factor's covariance w/ other
>> factors.  It would be 1 if the activity of a voxel were perfectly predictable
>> from that factor, and 0 if that factor had no power to predict it.  So, the
>> reason for having it is in the _interpretation_ of the statistic.
>>
>> Typically, it is computed as:
>>
>> Beta = b * (sx / sy)
>>
>> where
>>
>> Beta: standardized regression coefficient (-1 to 1)
>>
>> b: the unstandardized regression coefficient (can take any value), which is
>> the 'regulular' weight, sometimes confusingly referred to as beta but isn't
>> standardized) - probably a pe, which is equivalent to one of my copes in this
>> case
>>
>> sx: standard deviation of the EV
>> xy: standard deviation of the data
>>
>> Does that make any sense?
>>
>> Ed
>>
>>

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